The sample variance (commonly written or sometimes ) is the second sample central moment and is defined by

(1)

where the sample mean and is the sample size.

To estimate the population variance from a sample of elements with a priori unknown mean (i.e., the mean is estimated from the sample itself), we need an unbiased estimator for . This estimator is given by k-statistic , which is defined by

(2)

(Kenney and Keeping 1951, p. 189). Similarly, if samples are taken from a distribution with underlying central moments , then the expected value of the observed sample variance is

(3)

Note that some authors (e.g., Zwillinger 1995, p. 603) prefer the definition

(4)

since this makes the sample variance an unbiased estimator for the population variance. The distinction between and is a common source of confusion, and extreme care should be exercised when consulting the literature to determine which convention is in use, especially since the uninformative notation is commonly used for both. The unbiased sample variance is implemented as Variance[list].

Also note that, in general, is not an unbiased estimator of the standard deviation even if is an unbiased estimator for .

Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, p. 16, 2000.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

CITE THIS AS:

Weisstein, Eric W. "Sample Variance." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SampleVariance.html